Let $\Theta$ function be the function associated to a lattice $\Lambda=\oplus_{i\leq 2}Z\lambda_i\subset C$ of $C$ with transformation property defined as $\lambda\in\Lambda, \Theta(z+\lambda)=\Theta(z)e^{F(z,\lambda)}$ where $F$ is a linear polynomial in $z$ only.(Theta vanishes at points of $C$ periodically, though it is not periodic.) Let $\sigma$ be the Weierstraß sigma function associated to the lattice.
If $\Theta$ vanishes at $P_i$ with multiplicity $n_i$, then $g=\frac{\Theta(z)}{\sigma(z-P_i)^{n_i}}$ is a new theta function. So $g$ is trivial theta function.(i.e. $g=e^{a+bz+cz^2}$ for some $a,b,c\in C$.)
$\textbf{Q:}$ Book says any theta function can be expressed in terms of Weierstraß sigma function. How do I write trivial theta function in terms of Weierstraß sigma function? Trivial theta does not vanish and even better it is holomorphic. However Weierstraß sigma function has zeroes periodically on the lattice points and I can shift its zeros. I do not see any reason to write any theta function in terms of Weierstraß sigma function. I can say at best $\Theta(z)=e^{a+bz+cz^2}\prod_i\sigma(z-P_i)^{n_i}$ where $P_i$ are zeroes of multiplicity $n_i$ within a fundamental domain.
Ref: Analytic Theory of Abelian Varieties by Swinnerton-Dyer pg 20.